Fractional elasticity and Eringen's model are just a kernel apart
A Evgrafov, JC Bellido
Nonlocal elasticity theories have a long history in continuum mechanics. Their common objective is to take into account long-range internal interaction forces between particles, which nowadays becomes even more important owing to the continuing miniaturization of mechanical devices. These theories aim to alleviate a variety of singularity problems that arise in classical, local models, such as for example stress singularities in the vicinity of cracks. The beginning of non-local theory of elasticity goes back to the pioneering work of Kröner [1].
Arguably the most popular theory of non-local elasticity is due to Eringen [2]. This model has been utilized in a variety of mechanical applications, and it has recently attracted revitalized interest owing to its applicability to the modelling of nanobeams and nanobars [3]. In spite of such an interest in this model from the point of view of applications, even the most basic mathematical questions such as the existence and uniqueness of solutions have been rarely considered in the research literature for this model [4]. We will discuss precisely these questions, illustrating that the model is in general ill-posed in the case of smooth integral kernels, the case which appears rather often in numerical studies. We also consider the case of singular, non-smooth kernels and for the paradigmatic case of Riesz potential we establish the well-posedness of the model in fractional Sobolev spaces. For such a kernel, in dimension one the model reduces to the well-known fractional Laplacian (see for example the survey of applications [5] and the references therein), thereby providing a natural link between Eringen's integral model and fractional partial differential equations in the context of linear elasticity. Our development requires new ideas and tools, such as for instance a non-local version of Korn’s inequality.
Time permitting, we will also discuss a surprisingly non-trivial question of possible extensions of Eringen’s model with Riesz kernel to spatially heterogeneous material distributions, which is fuelled by our research interests in topology optimization.
[1] Kröner, E. Elasticity theory of materials with long range cohesive forces. Int J Solids Struct 1967; 3(5): 731–742.
[2] Eringen, AC. Nonlocal continuum field theories. New York: Springer Science & Business Media, 2002.
[3] Romano, G, Barretta, R, Diaco, M. On nonlocal integral models for elastic nano-beams. Int J Mech Sci 2017; 131: 490–499.
[4] Altan, S. Existence in nonlocal elasticity. Arch Mech 1989; 41(1): 25–36.
[5] Vázquez, JL. Recent progress in the theory of nonlinear diffusion with fractional Laplacian operators. Discrete Contin Dyn Syst Ser S 2014; 7(4): 857–885.